Brittany Terese Fasy is a researcher in topological data analysis. She started a research group TDA at MSU in Fall 2015. The group meets weekly to discuss fundamental topics, recent research papers, and current research of group members. More information on specific research projects of Dr. Fasy and her collaborators can be found below:
Statistical Computational Topology
Persistent homology is a method for probing topological properties of point clouds and function. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be “topological noise.” I am interested in bringing statistical ideas to persistent homology in order to distinguish topological signal from topological noise and to derive meaningful, yet computable, summaries of large datasets. For more information, please see the CMU TopStat website.
publications and preprints
 "Statistical Inference For Persistent Homology: Confidence Sets For Persistence Diagrams"
by
Brittany Terese Fasy,
Fabrizio Lecci,
Alessandro Rinaldo,
Larry Wasserman,
Sivaraman Balakrishnan, and
Aarti Singh.
To appear: Annals of Statistics.
[arXiv:1303.7117] [BibTeX]  "Stochastic Convergence of Persistence Landscapes and Silhouettes"
by Frédéric Chazal,
Brittany Terese Fasy,
Fabrizio Lecci,
Alessandro Rinaldo, and
Larry Wasserman.
Conference: Proceedings of SoCG 2014.
In submission: Journal of Computational Geometry.
[arXiv:1312.0308] [BibTeX]  "Subsampling Methods for Persistent Homology"
by Frédéric Chazal,
Brittany Terese Fasy,
Fabrizio Lecci,
Bertrand Michel,
Alessandro Rinaldo, and
Larry Wasserman.
Work in progress.
[arXiv:1406.1901] [BibTeX]  "On the Bootstrap for Persistence Diagrams and Landscapes"
by
Frédéric Chazal,
Brittany Terese Fasy,
Fabrizio Lecci,
Alessandro Rinaldo,
Aarti Singh,
and
Larry Wasserman.
Journal: Modeling and Analysis of Information Systems.
[arXiv:1311.0376] [BibTeX]
presentations and poster
 SoCG 2014: Stochastic Convergence of Persistence Landscapes and Silhouettes [Slides]
 JMM 2014: The Intersection of Statistics and Topology [Slides]
 Statistical Inference for Persistent Homology [Poster]
Map Construction and Comparison
In today's society, there is an overabundance of data. In particular, the movement of every cell phone carried by a person walking or every vehicle driven through a city can be traced by GPS and then recorded. Given the GPS trajectories of, for example, all taxis driving through a major city, I investigate how to synthesize the overabundance of information that these trajectories provide. One way of synthesizing the set of trajectories is to use them to estimate the underlying road network. This is called map (re)construction. I focus on evaluating map construction algorithms. Given two road networks, A and B, represented as graphs embedded in a compact subspace of R^2, the task is to compute a meaningful distance between the two road networks that takes into consideration both the local and the global differences. For example, A may be the ground truth road network and B may be the road network reconstructed from GPS trajectory data. Moreover, this research applies to more networks than just to road networks. For example, one may also consider biological networks, filaments of galaxies throughout the universe, and hiking paths.
publications and preprints
 "Local Persistent Homology Based Distance Between Maps" by
Mahmuda Ahmed,
BTF, and
Carola Wenk.
To appear (Nov. 2014): Proc. ACM SIGSPATIAL GIS.
[BibTeX] 
"New Techniques in Road Network Comparison" by
Mahmuda Ahmed,
BTF, and
Carola Wenk. Abstract.
Proc. Grace Hopper Celebration, Oct 2014.
[BibTeX] 
"PathBased Distance for Street Map Comparison" by
Mahmuda Ahmed,
BTF,
Kyle S. Hickmann,
and
Carola Wenk.
In submission: Transactions on Spatial Algorithms and Systems [axXiv:1309.6131] [BibTeX]
presentations
 Local Homology Based Distance [Slides]
Other
Modes of Gaussian Mixtures.
To many researcher's surprise,
n+1 isotropic Gaussian kernels in R^n can have n+2 modes. I call
these additional modes "ghost modes."
More generally, there exist finite configurations of isotropic
Gaussian kernels with superlinearly
many modes. Moreover, even the most simple configuration exhibits an
exponential number of critical points.
Curves in Space.
Researchers across many fields are interested in the topological
and geometric properties of shapes in Euclidean space.
In a 2011 Journal paper, I bound the difference of lengths of curves
in Euclidean space by a function of the total curvature and the Fréchet
distance between the curves.
Heat Equation Homotopy.
Persistence homology is a tool used to classify topological features that are present in data sets and functions.
I am interested in using persistence to explore the deep structure, or scale space, of images. The scale space
of an image is a family of related images obtained through convolution
with the Gaussian kernel. Viewing each image as a realvalued function, I stack
the corresponding persistence diagrams. This creates
a vineyard of curves that connect the points in the diagrams.
I am interested in using the vineyard arising from the heat equation homotopy
to define a distance between two images.
Computer Science Education.
I created virtual worlds using a 3D interactive animation environment,
Alice. Alice had
previously been used as a program visualization tool for introductory computer
science classes. The
worlds I created enable use of this tool at the intermediate programming level by
introducing the concepts of lists and arrays. This research
has provoked interest in expanding the use of Alice to higher level CS
courses.
Homotopy Classification Problems.
A basic problem in mathematics is the classification problem. In group theory we have the Classification Problem for Groups: Given a collection G of groups, classify the groups G in G up to isomorphism. And in homotopy theory, we have the Homotopy Classification Problem for Spaces: Given a collection T of topological spaces, classify the spaces X in T up to homotopy equivalence.
In some cases, these classification problems are actually equivalent. I am interested in understanding these cases.
In particular, I look at examples classifying groups by the centralizers and classifying the path components of function spaces.
publications and preprints

"Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions"
by Herbert Edelsbrunner,
BTF,
and Günter Rote.
Conference: SoCG 2012
Journal (2013): Discrete and Computational Geometry
[BibTeX] 
"The Difference of Lengths of Curves in R^n" by BTF.
Journal: Acta Sci. Math. (Szeged), 2011.
[BibTeX]  Prelim: "Discovering Metrics and Scale Space". 2010.
Unpublished manuscript: [PDF]
[BibTeX] 
Research Initiation Project: the Heat Equation Homotopy, 2009.
[axXiv:1002.1937] [BibTeX]  "Homotopy Classification of the Components of the Space of Maps into an Aspherical Space: a Problem in the intersection of Group Theory and Topology."
University Scholar Senior Thesis.
[BibTeX]
presentations
 SoCG 2012: Ghost Modes in Gaussian Mixtures [Slides]
 Prelim 2010: Discovering Metrics and Scale Space [Slides]
 Expanding Alice [Slides]